Chapter 1: Problem 3
Regarding \(2 x=5\) as the equation \(2 x+0 y=5\) in two variables, find all solutions in parametric form.
Short Answer
Expert verified
Parametric form: \((x, y) = (\frac{5}{2}, t)\) where \(t\in \mathbb{R}\).
Step by step solution
01
Understand the given equation
We are given the equation \(2x = 5\). We can interpret it in two dimensions by rewriting it as \(2x + 0y = 5\). This implies there is no \(y\) term affecting the solution.
02
Solve for x
From the equation \(2x + 0y = 5\), we simplify the equation to find the value of \(x\). We do this by dividing both sides by 2: \(x = \frac{5}{2}\).
03
Express y in terms of a parameter
Since the \(y\) term has a coefficient of zero, \(y\) can be any real number. To express it parametrically, set \(y = t\) where \(t\) is a parameter that can take any real value.
04
Write the solution in parametric form
Combine our findings: \(x = \frac{5}{2}\) and \(y = t\). Therefore, the solution in parametric form is \((x, y) = (\frac{5}{2}, t)\), where \(t\in \mathbb{R}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Equations
Linear equations are foundational to algebra, serving as equations of straight lines when graphed. A linear equation in two variables is typically written in the form \[ ax + by = c, \]where \(a\), \(b\), and \(c\) are constants.
- The equation represents a line on a 2D plane.
- In the exercise above, \(2x + 0y = 5\) is a type of linear equation. Here, \(a = 2\) and \(b = 0\).
- Such equations can be solved to find the values of \(x\) and \(y\) that satisfy the equation.
Solution in Two Variables
The solution of linear equations in two variables involves finding a set of values for \(x\) and \(y\) that satisfy the equation.
To solve it fully, we can resolve the equation into its simpler form: \(x = \frac{5}{2}\),giving us a constant value for \(x\).
On the other hand, because \(y\) can be any number, the complete set of solutions is a vertical line where all \(y\) values are acceptable parallel to the y-axis.
- When an equation has two variables, it depicts a relationship between them.
- The solution is typically a collection of all point pairs \((x, y)\) lying on the graph of the equation.
To solve it fully, we can resolve the equation into its simpler form: \(x = \frac{5}{2}\),giving us a constant value for \(x\).
On the other hand, because \(y\) can be any number, the complete set of solutions is a vertical line where all \(y\) values are acceptable parallel to the y-axis.
- This means, for every value of \(y\), the equation maintains its truth, resulting in an infinite number of solutions.
Parametric Form Solution
The parametric form is an excellent way to express the solutions of an equation by using parameters.
In parametric equations, each coordinate is expressed as a function of one or more parameters.
The parametric form becomes \((x, y) = (\frac{5}{2}, t)\), where \(t\) can represent any real number.
In parametric equations, each coordinate is expressed as a function of one or more parameters.
- For the given linear equation, the parameter \(t\) stands for a possible range of values for \(y\).
- This allows us to write equations as functions of \(t\), a real number.
The parametric form becomes \((x, y) = (\frac{5}{2}, t)\), where \(t\) can represent any real number.
- This form assures us that \(x\) remains constant while \(y\) is free to vary, depicting a vertical line in the graph.
- The parametric form is beneficial for understanding the flexibility in solutions where certain variables don't impact or limit the others' values.